In the situration shown in the figure the friction coefficient between `M` and the horizontal surface is `mu`. The force `F` is applied at an angle `theta` with vertical. The correct statements are

Options

A. If `theta gt tan^(-1) mu` the block cannot be pushed forward for any value of F
B. If `theta lt tan^(-1) mu` the block cannot be pushed forward for any value of F
C. As `theta` decreases the magnitude of force needed to just push the block M forward increases .
D. none of the above

Answer

A, C

Explanation

N = F cos `theta + Mg " " …..(i)`
`f_("max") = (mu(M -2m))/(3) " " …. (ii) `
To just push the block F sin `theta = f_("max")`
`implies F sin theta mu ( F cos theta + Mg)`
`implies F = (mu Mg)/(sin theta - mu cos theta )`
`implies sin theta - mu cos theta gt 0 implies tan theta gt mu `
`implies tan theta gt tan (tan^(-1) mu) implies theta gt tan ^(-1) mu `
Hence , the block can be pushed forward only if `theta gt tan^(-1) mu`
Again as `theta` decreases sin `theta` decreases while cos `theta` increases , therefore , sin `theta - mu cos theta` decreases . Hence , `(mu Mg)/(sin theta - mu cos theta)` increases .